Affine maximal hypersurfaces are extremals of the interior variation of the affinely invariant volume. The corresponding Huler-Lagrange equation is a fourth order PDE ( see (8) below ). Oringinally, these hypersurfaces are called " affine minimal hypersurfaces". Calabi calculated the second variation and proposed to call them " affine maximal". On affine maximal surfaces S.S.Chern made the following conjecture (se[CH]), which is called the affine Bernstein problem: Conjecture : Let x3 =

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... be a strictly conver function defined for all (x1,22) € A?. If M = {(£1, %2, f(£1,£2))} is an affine mazimal surface, then M must be an elliptic paraboloid. This conjecture was generalized to higher dimensions. This is a long standing problem. Some partial solutions to this conjecture have been made by several authors (see [CA], [L1],[P2]).In this paper we solve this conjecture for 2 dimension and 3 dimension. Precisely, we will prove the following Theorem: Let tn41 = f(£1,.-.,2n) be @ strictly conver function defined for all (x1,...,%,) € A". Suppose that M = {(21,...,2n,f(1,...,2n))} ts an affine maximal hyersurface. Then in the case n= 2 orn=3, M must be an elliptic paraboloid. After the main part of our proof was completed we learned that Trudinger and Wang X. J also claimed that they solved this problem for 2-dimension, using different method.